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Characterization of Global Fields by Dirichlet L-Series

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Characterization of Global Fields by Dirichlet L-Series Cornelissen et al. Res. Number Theory (2019) 5:7 https://doi.org/10.1007/s40993-018-0143-9 RESEARCH Characterization of global fields by Dirichlet L-series Gunther Cornelissen1,BartdeSmit2,XinLi3, Matilde Marcolli4,5,6 and Harry Smit1* *Correspondence: [email protected] Abstract 1Mathematisch Instituut, Universiteit Utrecht, Postbus We prove that two global fields are isomorphic if and only if there is an isomorphism of 80.010, 3508 TA Utrecht, The groups of Dirichlet characters that preserves L-series. Netherlands Keywords: L Full list of author information is Class field theory, -series, Arithmetic equivalence available at the end of the article Mathematics Subject Classification: 11R37, 11R42, 11R56, 14H30 Part of this work was done whilst the first and third author enjoyed the hospitality of the University of Warwick (special thanks to Richard Sharp for making it 1 Introduction possible). As was discovered by Gaßmann in 1926 [10], number fields are not uniquely determined up to isomorphism by their zeta functions. A theorem of Tate [20] from 1966 implies the same for global function fields. At the other end of the spectrum, results of Neukirch and Uchida [15,21,22] around 1970 state that the absolute Galois group does uniquely determine a global field. The better understood abelianized Galois group again does not determine the field up to isomorphism at all, as follows from the description of its character group by Kubota in 1957 ([13], compare [1,17]). Funakura ([9,§I],using[8, Thm. 5]) has shown in 1980 that there exists a number field k (of degree 255! over Q) and two non- isomorphic abelian extensions K/k and L/k of degree 4 with a bijection between all Artin L-series of K/k and L/k (or, equivalently, between the L-series of the four occurring characters). In this paper, we prove that two global fields K and L are isomorphic if and only if there q q ab ∼ q ab exists an isomorphism of groups of Dirichlet characters ψ : GK −→ GL that preserves q q ab L-series: LK(χ) = LL(ψ(χ)) for all χ ∈ GK . A more detailed series of equivalences can be found in the Main Theorem 3.1 below. To connect this theorem to the discussion in the previous paragraph, observe that the existence of ψq without equality of L-series is the same as K and L having abelianized Galois groups that are isomorphic as topological groups, and that for the trivial character χtriv,wehaveLK(χtriv) = ζK, so that preserving L-series at χtriv is the same as K and L having the same zeta function. Contrary to the result of Funakura, our characters do not factor over a fixed Galois group. In global function fields we explicitly construct the isomorphism of function fields via a map of kernels of reciprocity maps (much akin to the final step in Uchida’s proof [22]). For number fields, we do not need the full hypothesis: we can prove the stronger result that for every number field, there exists a character of any chosen order > 2 for which the L-series does not equal any other Dirichlet L-series of any other field (see Theorem 10.1). © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, 123 provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 0123456789().,–: volV 7 Page 2 of 15 Cornelissen et al. Res. Number Theory (2019) 5:7 The method here is not via the kernel of the reciprocity map, but rather via representation theory. We briefly indicate the relation of our work to previous results. The main result was first stated in a 2010 preprint by two of the current authors, who discovered it through methods from mathematical physics ([4], now split into two parts [5]and[6]). For function fields, the main result was proven by one of the authors in [3], using dynamical systems, referring, however, to [4] for some auxiliary results, of which we present the first published proofs in the current paper. Sections 9 and 10 contain an independent proof for the number field case that was found by the second author in 2011. The current paper not only presents full and simplified proofs, but also uses only classical methods from number theory (class field theory, Chebotarev, Grunwald-Wang, and inverse Galois theory). In [19], reconstruction of field extensions of a fixed rational function field from relative L-series is treated from the point of view of the method in Sects. 9 and 10 of this paper. After some preliminaries in the first section, we state the main result in Sect. 3.The next few sections outline the proof of the various equivalences in the main theorem, using basic class field theory and work of Uchida and Hoshi. In the final sections, we deal with number fields, and use representation theory to prove some stronger results. 2 Preliminaries In this section, we set notation and introduce the main object of study. A monoid is a semigroup with identity element. If R is a ring, we let R∗ denote its group of invertible elements. Given a global field K,weusethewordprime to denote a non-zero prime ideal if K is a number field, and to denote an irreducible effective divisor if K is a global function field. Let PK be the set of primes of K.Ifp ∈ PK,letvp be the normalized (additive) valuation corresponding to p, Kp the local field at p,andOp its ring of integers. Let AK,f be the finite ∗ K OK A adele ring of , its ring of finite integral adeles and K,f the group of ideles (invertible finite adeles), all with their usual topology. Note that in the function field case, infinite placesdonotexist,sothatinthatcase,AK,f = AK is the adele ring, OK is the ring of A∗ = A∗ K integral adeles and K,f K is the full group of ideles. If is a number field, we denote by OK its ring of integers. If K is a global function field, we denote by q the cardinality of the constant field. Let IK be the multiplicative monoid of non-zero integral ideals/effective divisors of our K P m ∈ global field ,soIK is generated by K. We extend the valuation to ideals: if IK p ∈ P m ∈ Z m = pvp(m) and K,wedefinevp( ) ≥0 by requiring p∈P K .LetN be the norm function on the monoid IK: it is the multiplicative function defined on primes p by p N(p):= #OK/p if K is a number field and N(p) = qdeg( ) if K is a function field. Given two global fields K and L, we call a monoid homomorphism ϕ : IK → IL norm-preserving if N(ϕ(m)) = N(m) for all m ∈ IK. We have a surjective monoid homomorphism · A∗ ∩ O → → pvp(xp) ( )K : K,f K IK, (xp)p , p a restriction of the usual homomorphism from ideles to fractional ideals—we use the monoid version in Sect. 6.1 and in the companion paper [6]. A split for (·)K is by definition ∗ K K → A ∩ OK p K p = a monoid homomorphism s : I K,f such that for every prime , s ( ) ... π ... π ∈ K · ◦ = ( , 1, p, 1, ) for some uniformizer p p. It follows that ( )K sK idIK . Cornelissen et al. Res. Number Theory (2019) 5:7 Page 3 of 15 7 ab ab Let GK be the Galois group of a maximal abelian extension K of K, a profinite topo- ∗ ab logical group. There is an Artin reciprocity map AK → GK . In the number field case, we A∗ A∗ A∗ → ∈ A∗ embed K,f into the group of ideles K via K,f x (1,x) K, restrict the Artin ∗ A K K reciprocity map to K,f and call this restriction rec . In the function field case, rec is just the (full) Artin reciprocity map. q ab ab Let T denote the unit circle, equipped with the usual topology, and GK = Hom (GK , T) ab q ab the group of continuous linear characters of GK . Given χ ∈ GK ,wewrite χ = p ∈ P χ| ∗ = U( ): K: recK(Op) 1 χ χ for the set of primes where is unramified .Wedenoteby U( ) the submonoid of IK χ m ∈ χ m = p ∈ P \ χ generated by U( ), i.e., IK is in U( ) if and only if vp( ) 0 for all K U( ). ∗ For m ∈ U(χ) , we set (in a well-defined way, as the choice of sK is up to an element of OK): χ(m):= χ(recK(sK(m))), and for m ∈ IK\ U(χ) we set χ(m) = 0. q ab ab For any χ ∈ GK , the kernel ker χ is an open subgroup of GK . Therefore, the fixed field of χ, denoted Kχ , is a finite abelian extension of K. Even more: as the extension is finite, there is an n such that χ n is the trivial character. It follows that im χ is a subgroup of ∼ the nth roots of unity, hence cyclic. As we have an isomorphism im χ −→ Gal(Kχ /K), we obtain that Kχ /K is a finite cyclic extension. Conversely, for any finite cyclic extension K q ab ab of K there exists a character χ ∈ GK such that ker χ = Gal(K /K ). The following two lemmas are easy, but we include a proof in the terminology of this paper for lack of a suitable reference.
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